Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois
representation with cyclotomic determinant, and let $N>1$ be an
integer that is square mod $p$. There exist two twisted modular
curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined
over~$\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves
of degree $N$ realizing $\varrho$. The paper focuses on the only
genus-three instance: the case $N\!=7,\,p=3$. From an explicit
description of the au...

Let $\varrho\colon G_\mathbb{Q}\longrightarrow PGL_2(\mathbb{F}_p)$ be a Galois
representation with cyclotomic determinant, and let $N>1$ be an
integer that is square mod $p$. There exist two twisted modular
curves $X^+(N,p)_\varrho$ and $X^+(N,p)'_\varrho$\, defined
over~$\mathbb{Q}$ whose rational points classify the quadratic $\mathbb{Q}$-curves
of degree $N$ realizing $\varrho$. The paper focuses on the only
genus-three instance: the case $N\!=7,\,p=3$. From an explicit
description of the automorphism group of the modular curve $X_0(63)$,
it follows that the twisted curves are isomorphic over $\mathbb{Q}$
in this case. We also obtain a plane quartic equation for the
twists and then produce the desired $\mathbb{Q}$-curves, provided that the
set of rational points on this quartic can be determined. The
existence of elliptic quotients and of an unramified double cover
$X(7,3)_\varrho$ having a genus-two quotient permits
a variety of combinations of covers and Prym-Chabauty methods to
determine these rational points. We include two examples where
these methods apply.